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**Variational formulation, discrete conservation laws, and path-domain independent integrals for elasto-viscoplasticity.**
*(English)*
Zbl 0739.73046

This paper is concerned with the development of variational principles and associated conservation laws for time-discrete plasticity and viscoplasticity. That is, the time interval \([0,T]\) is discretised into subintervals by \(0=t_ 0<t_ 1<\dots<t_ n=T\); the state is assumed known at time \(t_ n\), and the problem is to find the state at \(t_{n+1}\).

The authors begin by summarising classical plasticity and viscoplasticity; hardening is included. They then introduce the total energy at time \(t\) by means of the Hu-Washizu functional, and also the Lagrangian functional associated with plastic dissipation over the entire body up to time \(t\). The discrete Lagrangian is obtained by introducing an Euler backward approximation for rates. The expression for the discrete Lagrangian is then used to obtain an expression for the total energy, in the discrete formulation. The Euler-Lagrange equations associated with the total energy yield the governing equations for the problem, including the closest-point projection algorithm and the discrete Kuhn-Tucker conditions.

The authors next exploit these results to derive material conservation laws using Noether’s theorem. It is seen, though, that the discrete Lagrangian is not invariant with respect to the three-parameter group of translations, so that the associated Noether quantity (the analogue of the energy-momentum tensor) is not conserved. The divergence of this quantity can, however, be explicitly found for homogeneous materials. By using divergence theorem the authors derive an integral quantity, denoted \(J^*\), which is path-domain independent and which vanishes in the absence of singularities. This result is directly analogous to that obtained by Budiansky and Rice in the context of elasticity [e.g.: B. Budiansky and J. R. Rice, Wave motion 1, 187-192 (1979; Zbl 0423.73077); J. Appl. Mech. 40, 201-203 (1973; Zbl 0261.73059)]. It is shown, furthermore, that \(J^*\) is directly interpretable as an energy release rate.

This is a well-written paper with sufficient background detail and references to be accessible to the nonspecialist in variational princples.

The authors begin by summarising classical plasticity and viscoplasticity; hardening is included. They then introduce the total energy at time \(t\) by means of the Hu-Washizu functional, and also the Lagrangian functional associated with plastic dissipation over the entire body up to time \(t\). The discrete Lagrangian is obtained by introducing an Euler backward approximation for rates. The expression for the discrete Lagrangian is then used to obtain an expression for the total energy, in the discrete formulation. The Euler-Lagrange equations associated with the total energy yield the governing equations for the problem, including the closest-point projection algorithm and the discrete Kuhn-Tucker conditions.

The authors next exploit these results to derive material conservation laws using Noether’s theorem. It is seen, though, that the discrete Lagrangian is not invariant with respect to the three-parameter group of translations, so that the associated Noether quantity (the analogue of the energy-momentum tensor) is not conserved. The divergence of this quantity can, however, be explicitly found for homogeneous materials. By using divergence theorem the authors derive an integral quantity, denoted \(J^*\), which is path-domain independent and which vanishes in the absence of singularities. This result is directly analogous to that obtained by Budiansky and Rice in the context of elasticity [e.g.: B. Budiansky and J. R. Rice, Wave motion 1, 187-192 (1979; Zbl 0423.73077); J. Appl. Mech. 40, 201-203 (1973; Zbl 0261.73059)]. It is shown, furthermore, that \(J^*\) is directly interpretable as an energy release rate.

This is a well-written paper with sufficient background detail and references to be accessible to the nonspecialist in variational princples.

Reviewer: B.D.Reddy (Rondebosch)

### MSC:

74S30 | Other numerical methods in solid mechanics (MSC2010) |

74P10 | Optimization of other properties in solid mechanics |

74C10 | Small-strain, rate-dependent theories of plasticity (including theories of viscoplasticity) |

74A20 | Theory of constitutive functions in solid mechanics |